It's not simple and sometimes impossible to divide a circle into n equal parts. That is, if the word "circle" signifies the border of a circular region, then, as was shown by the great Gauss, the construction with compass and ruler is only possible when Euler function φ is a power of 2: φ(n) = 2n. This happens exactly when n is the product of Fermat primes and a power of 2. Here are a few such numbers 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24 ... In particular Euclidian construction of a regular heptagon, i.e. a polygon with 7 sides is impossible.

However, if the question is of dividing the area of a circle, not the circumference, the problem is solvable in an elegant way that works for any n > 1.

Here's an example of a solution for n = 7.

Divide a diameter of the circle into 7 equal parts. Now draw curves that consist of two semicircles: one above the diameter, the other below the diameter. The diameter of the left/upper semicircle takes up, say, k parts whilst the diameter of the right/lower part takes (7-k) parts.

Remarkably, the length of each of the curves equals the length of the original semicircle. More Importantly, each of the regions bounded by two successive curves is exactly 1/7 of the given circle.

Let's check this in a general case of n parts. Take two curves: one on, say, k and n-k parts, the other on m and n-m parts, m >k. Assuming each part equals 2 so that k, n-k, m, and n-m are the radii of the four semicircles, the area enclosed by the two curves is half the sum

π((m/2)² - (k/2)²) + π[((n - k)/2)² - ((n - m)/2)²]

This is equal to

π/8·2n(m - k).

In case m = k + 1, i.e., when the region at hand is bounded by two successive curves the area is

πn/4.

But, in our notations, the radius of the big original circle is n/2 and the area is

How would you go about dividing a pie into 6 equal area portion rings so each ring matches the area of the inner core. First divide the pie total area by 7 then 1/7th is core area. How then to proceed for each of the 6 outer rings?